Special coordinate coverings of Riemann surfaces

Gunning, Robert C.

doi:10.1007/bf01362287

Cited by 121 publications

(123 citation statements)

References 8 publications

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“…Similar straightforward computations show λ i0 0 = 2γ i , λ 0i 0 = 0, and λ i[jk] = γ i ω jk , from which follows λ iα α = λ αi α = 2nγ i . This justifies the second equality of (9).…”

Section: This Shows Thatsupporting

confidence: 74%

“…From this there follows that S ij k = λ ij k −Ξ ij k = λ (ij) k −Ξ (ij) k . Because S ij k must be completely trace free, there follows λ (ip) p = (2n − 1)γ i , and this implies the first equality of (9). Similar straightforward computations show λ i0 0 = 2γ i , λ 0i 0 = 0, and λ i[jk] = γ i ω jk , from which follows λ iα α = λ αi α = 2nγ i .…”

Section: This Shows Thatmentioning

confidence: 74%

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Contact Schwarzian Derivatives

Fox

2005

Nagoya Mathematical Journal

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Abstract. H. Sato introduced a Schwarzian derivative of a contactomorphism of R 3 and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a contact projective structure is defined to be a cocycle of the contactomorphism group taking values in the space of sections of a certain vector bundle associated to the contact structure, and measuring the extent to which a contactomorphism fails to be an automorphism of the contact projective structure. For the flat model contact projective structure, this gives a contact Schwarzian derivative associating to a contactomorphism of R 2n−1 a tensor which vanishes if and only if the given contactomorphism is an element of the linear symplectic group acting by linear fractional transformation. §1. IntroductionThe classical Schwarzian derivative is a cocycle, S(f ), of the diffeomorphism group of the real line with coefficients in the quadratic differentials and which vanishes when restricted to the group of projective transformations. It arises naturally in the context of flat projective structures on one-dimensional manifolds and may be interpreted as describing how a normalized second order linear differential operator transforms under a change of variable, provided the operator is viewed as acting on −1/2 densities, rather than on functions (this determines the relevant normalization). Its characteristic properties are:

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Section: This Shows Thatsupporting

confidence: 74%

Section: This Shows Thatmentioning

confidence: 74%

Contact Schwarzian Derivatives

Fox

2005

Nagoya Mathematical Journal

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“…It is easy to see that the derivative troubles can be overcome by inverting equation (2.5) iteratively, 10) and where each of the b (r) j for j ≥ 2 is a polynomial expression in the a (s) k , namely for…”

Section: Behaviour Under Finite Holomorphic Change Of Complex Coordinmentioning

confidence: 99%

“…In this paper we discuss the former aspect in the framework of the W algebras [1], realized over Riemann surfaces [2] - [4]. The intricacies encountered are well known and can be found in the literature [5] - [9], and the outstanding mathematical books [10] as well.…”

Section: Introductionmentioning

confidence: 99%

“…The main difficulty lies in the definition of differential operators on Riemann surfaces [5]; in particular their equipment in a "conformal covariant" frame has been widely discussed in the literature [3], [10] [45]. In particular the more general question of classifying all holomorphically covariant differentials operators amounts to the study of operators which are covariant with respect to projective transformations.…”

Section: Introductionmentioning

confidence: 99%

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Primary currents and Riemannian geometry in algebras

Bandelloni

Lazzarini²

2001

Nuclear Physics B

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Complex projective structures with maximal number of Möbius transformations

Gianluca

Ruffoni

2018

Mathematische Nachrichten

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We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More generally we show that Galois Belyĭ curves are precisely those Riemann surfaces for which the Fuchsian uniformization is the unique complex projective structure invariant under the full group of biholomorphisms.

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Special coordinate coverings of Riemann surfaces

Cited by 121 publications

References 8 publications

Contact Schwarzian Derivatives

Contact Schwarzian Derivatives

Primary currents and Riemannian geometry in algebras

Complex projective structures with maximal number of Möbius transformations

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